Solve 1/3k(k^2-1)+(k+1)((k+1)-1))=1/3(k+1)((k+1)^2-1)

$$\frac { 1 } { 3 } k ( k ^ { 2 } - 1 ) + ( k + 1 ) ( ( k + 1 ) - 1 ) ) = \frac { 1 } { 3 } ( k + 1 ) ( ( k + 1 ) ^ { 2 } - 1 )$$

Evaluate

$$\frac{k\left(k+1\right)\left(k+2\right)}{3}$$

Solution Steps

Use the distributive property to multiply \frac{1}{3}k by k^{2}-1.

$$\frac{1}{3}k^{3}-\frac{1}{3}k+\left(k+1\right)\left(k+1-1\right)$$

Subtract 1 from 1 to get 0.

$$\frac{1}{3}k^{3}-\frac{1}{3}k+\left(k+1\right)k$$

Use the distributive property to multiply k+1 by k.

$$\frac{1}{3}k^{3}-\frac{1}{3}k+k^{2}+k$$

Combine -\frac{1}{3}k and k to get \frac{2}{3}k.

$$\frac{1}{3}k^{3}+\frac{2}{3}k+k^{2}$$

Expand

$$\frac{k^{3}}{3}+k^{2}+\frac{2k}{3}$$

Solution Steps

Use the distributive property to multiply \frac{1}{3}k by k^{2}-1.

$$\frac{1}{3}k^{3}-\frac{1}{3}k+\left(k+1\right)\left(k+1-1\right)$$

Subtract 1 from 1 to get 0.

$$\frac{1}{3}k^{3}-\frac{1}{3}k+\left(k+1\right)k$$

Use the distributive property to multiply k+1 by k.

$$\frac{1}{3}k^{3}-\frac{1}{3}k+k^{2}+k$$

Combine -\frac{1}{3}k and k to get \frac{2}{3}k.

$$\frac{1}{3}k^{3}+\frac{2}{3}k+k^{2}$$